For thirty years, two-dimensional systems of strongly correlated fermions are the subject of intense research in condensed matter ( high critical temperature superconductors , 3He films , quantum Hall effect, etc. . ) . Some of these systems , known as the spin liquids exhibit non-conventional magnetic properties . Our work focuses on the study of exotic orders characterizing these spin liquids , as well as neighboring models ( dimers and quantum partitions) , and are guided by both experiments on many compounds ( frustrated on the triangular lattice or grid systems kagome , nematic , spin ladders and tubes , etc. . ) and intimate links with quantum information ( entanglement, topological order , etc. . ) . In particular we study the structural changes of the ground states responsible for quantum phase transitions ( at zero temperature ) and the excitations which are often fractional . This phenomenon is the cause fractionalization of quantum statistics anyoniques quasiparticles ( present only two dimensions ) for the critical topological quantum computation . We also develop , more recently , interactions with experimentalists to elucidate the properties of liquid type spins new materials ( kapellasite , herbersmithite , etc. . ) . The tools used to carry out these various studies are not only numerical (exact diagonalizations , simulations, variational approaches) but also analytical and semi- analytical ( perturbative continuous unitary transformations , effective theories of low energy, high temperature series).

Dense phase gas to form an electron Fermi liquid , as is the case for example of the valence electrons of the solid sodium . The low energy excitations are characterized by the effective mass of quasi- particles, their spectral weight Z , etc. . In less dense phase , the electron gas is more correlated and eventually form a Wigner crystal . We are interested in the quantitative description of the ground state and excitations of gas depending on its density, in two and three dimensions. Are there different exotic phases of Fermi liquid and Wigner crystal ? In the framework of the Hartree-Fock approximation (for a homogeneous electron gas ), we find a richer than first order transition between liquid and crystal scenario: the electron density in the ground state is always periodic and varies continuously between the crystal and the Fermi liquid , resulting in densities of anisotropic states . Whether or not these conditions persist in the presence of correlations is one of our main objectives. We also want to determine the parameters of Landau and study the quasi- two-dimensional electron gas and the influence of impurities .

Ultracold atomic gases provide an experimental realization of strongly correlated quantum fluids and interest thereby theorists condensed matter. These systems are characterized by a remarkable control of the experimental parameters and the possibility of a quantitative comparison between theory and experiment . They not only allow the simulation of model Hamiltonians solids ( fermionic gas in reduced dimensions, quantum particles moving on a network, etc. . ) But also achieving unparalleled systems in condensed matter "traditional" (mixtures fermions - bosons , bosons or fermions with high spin quantum number , etc. . ) . Part of our business is the study of systems with a small number of bodies in the context of low energy effective approaches ( study of bound states of Efimov three or four bodies , etc. . ) . Another part of the study of the collective properties of low energy ultracold gas from numerical methods (Monte Carlo quantum ) or semi- analytical ( effective theories of low energy group of non- perturbative renormalization ) and focuses on three main areas: i) the fermionic gas -dimensional multi-component spin ( phase diagram , exotic superfluid phases , confinement and analogy with QCD) , ii) two-dimensional gas of bosons ( thermodynamics and transition of Kosterlitz - Thouless ) iii ) the superfluid -insulator Mott transition of a gas of bosons in two or three dimensional optical network ( quantum criticality and universality).

Interacting quantum systems have generically tend to entangle . The study of entanglement and its consequences is therefore of great interest . Our business is growing along two axes : (1) the entanglement of qubits , for which we are working on a detailed description of the Hilbert space , under an appropriate measure of the entanglement, in particular by using geometric tools , Hopf fibrations , Moebius transformations . We also plan , in the topological quantum computation , analysis by the intricacy of anyons interaction . (2 ) the consequences of entanglement with the environment. For small quantum systems , the environment can not be ignored. Under the influence of the latter, the system state does not remain pure ( decoherence ) and generally tends towards a steady state. We study (i ) the influence on the evolution of the characteristics of the environment ( thermodynamic phase ... ), (ii ) steady-state non-equilibrium system when the environment is non-equilibrium , ( iii ) phenomena existing relaxation in an isolated system .

Theme 5: Combinatorics of quantum collective phenomena (KA Penson)

The objective of this program is the development and application of methods of combinatorial analysis in the theory of condensed matter. It is well known that combinatorial analysis is at the heart of statistical mechanics . Demonstrations of quantum statistics of Fermi- Dirac and Bose -Einstein are using the enumeration of combinatorial quantum states in mind , thereby obtaining their explicit forms . In addition, in the other chapters of statistical mechanics and quantum optics on a large number of combinatorial problems meeting. We obtained a series of exact solutions of problems in Bose statistics , which relate to the normal order of operators. Interesting combinatorial sequences ( Bell numbers , Stirling , Catalan and their generalizations ) will appear in a natural way . Combinatorial models of quantum field theory (C. Bender et al. ) Are based on the knowledge of various combinatorial constructions which require the introduction of Hopf algebras .